Computelet美人2

(1)LetAnbetherealn*nmatrix(n≥2)whoseentryinposition(i,j)isij.
WhatistherankofAnasafunctionofn2(2)Letp(x)bethepolynomialleftasremainderwhenx20191isdividedbyx6+1.
Whatistheremainderleftwhenp(x)isdividedbyx326(3)LetSbethesetofall(unordered)pairsofdistincttwodigitintegers(intheusualdecimalnotation).
Ifamember{a,b}ofSispickedatrandom,whatistheprobabilitythata+biseven44/89(4)Fornapositiveinteger,letfn(x)bethecontinuousfunction1/(1+nx)withdomainthepositiverealnum-bers.
Letf(x)bethepointwiselimitofthesequence{fn(x)}n≥1offunctions.
Onwhichofthefollowingintervalsistheconvergencefn→funiformChooseallthecorrectoptions:(b),(c)(a)(0,1)(b)(1,2)(c)(2,∞)(d)Noneoftheabove.
(5)Putθ:=π/2019andletNdenotethesetofpositiveintegers.
WhichofthefollowingsubsetsofthereallineiscompactChooseallthecorrectoptions:(a),(c)(a){sinnθn|n∈N}(b){cosnθn|n∈N}(c){tannθn|n∈N}(d)Noneoftheabove.
(6)Thesmallest(positive)integerwithexactly20divisors(including1anditself)is:240(E.
g.
,10hasexactlyfourdivisors,namely,1,2,5,and10.
)(7)HowmanygrouphomomorphismsaretherefromZ/3Z*Z/4Z*Z/9ZtoZ/18Z54HereZ/nZdenotesthecyclicgroupofordern,andA*BthecartesianproductofAandB.
(8)Let,fortarealnumber,tdenotethelargestintegernotlargerthant.
Compute100.
50.
75f(t)dtforf(t):=tt12.
1/32or0.
03125(9)LetMdenotethereal6*6matrixallofwhoseoff-diagonalentriesare1andallofwhosediagonalentriesare5.
ListouttheeigenvaluesofM(eacheigenvaluemustbewrittenasmanytimesasitsmultiplicity):6,6,6,6,6,0(10)LetSbethesetofall2*3realmatriceseachofwhoseentriesis1,0,or1.
(Thereare36matricesinS.
)RecallthatthecolumnspaceofamatrixMinSisthesubspaceofR2(thevectorspaceof2*1realmatrices)spannedbythethreecolumnsofM.
FortwoelementsMandMinS,letuswriteMMifMandMhavethesamecolumnspace.
Notethatisanequivalencerelation.
HowmanyequivalenceclassesarethereinS6(11)Giventhatf(x,y)=u(x,y)+iv(x,y)isanentirefunctionofz=x+iysuchthatf(0)=1,u/x=(ey+ey)cosx,andu/y=(eyey)sinx,whatisf(π/3)312(12)LetA:=Z/6Zbethegroupofresidueclassesmodulo6ofintegers.
LetGbethegroupofbijections(asaset)ofA,themultiplicationbeingcomposition.
LetHbethesubgroupofGconsistingofthosebijectionsσsuchthatσ(x+2)=σ(x)+2forallxinA.
WhatistheindexofHinG40(13)LetS:={xaninteger|99ThesumoftheelementsofSis:7920.
(14)LetAbe4*5realmatrix.
ConsiderthesystemAx=boflinearequationswherexisa5*1columnmatrixofindeterminatesandbissomexed4*1columnmatrixwithrealentries.
GiventhatAisrowequivalenttothematrixRbelow(whichmeansthattherowsofAarealllinearcombinationsoftherowsofRandviceversa),andcanddbelowarebothsolutionstoAx=b,whatisthevalueofy7R=12130000010000000000,c=12345,d=y3455(15)Letnbetheleastpositiveintegersuchthat2≤k≤n1k≥5.
Choosethecorrectoption:(c)(a)n≤32(b)32(17)LetAbeareal2*2matrixsuchthatA6=I(whereIdenotestheidentity2*2matrix).
ThetotalnumberofpossibilitiesforthecharacteristicpolynomialofAis:5(18)TheshortestdistancefromtheorigininR3tothesurfacez2(x1)(y1)=2is24/3or8/3.
(19)Forrapositiverealnumberletf(r):=Crsinzzdz,whereCristhecontourreiθ,0≤θ≤π.
Whatislimr→0f(r)r2(20)Adegree3polynomialf(x)withrealcoefcientssatisesf(1)=2,f(2)=2,f(2)=2,andf(2)=12,wheref(x),f(x),andf(x)aretherst,second,andthirdderivativesoff(x)respectively.
Whatisf(2)5(21)Considerthefunctionf(z)=z+2z2+3z3+···=n≥0nzndenedontheopendisk{z||z|<1}.
Choosethecorrectoption:(b)(a)fisnotinjectivebutattainseverycomplexvalueatleastonce.
(b)fisinjectivebutdoesnotattaineverycomplexvalue.
(c)fisinjectiveandattainseverycomplexvalue.
(d)Noneoftheabove.
(22)Findtheareaoftheregion{(x,y)|0≤x,0≤y,x2/3+y2/3≤1}.
3πππ/323(23)Thenumberofsolutions(x,y)∈Z*Z,whereZdenotesthesetofintegers,oftheequationx2+16=y2is6.
(24)Evaluatelimn→∞(11n+1n21n3)n:1/e(25)Letkbetheeldwithexactly7elements.
LetMbethesetofall2*2matriceswithentriesink.
HowmanyelementsofMaresimilartothefollowingmatrix560001(26)Forthefunctionf(x)onthereallineRdenedbelow,whichofthefollowingstatementsaboutfistrueChooseallthecorrectoptions:(b),(c)f(x):=n≥1sin(x/n)n(a)fiscontinuousbutnotuniformlycontinuousonR.
(b)fisuniformlycontinuousonR.
(c)fisdifferentiableonR.
(d)fisanincreasingfunctiononR.
(27)LetfbeacontinuousfunctionfromthereallineRtotheclosedinterval[1,3]suchthat:f1(1)andf1(3)aresingletons,andf1(x)consistsofexactlytworealnumbersforeveryxin(1,2)∪(2,3).
Whichofthefollowingcanbethecardinalityoff1(2)Chooseallthecorrectoptions:(a),(d)(a)1(b)2(c)countableinnity(d)uncountableinnity(28)WhichofthefollowingfunctionsisuniformlycontinuousonthegivendomainChooseallthecorrectoptions:(a),(d)(a)1/x2on[1,∞).
(b)1/xon(0,∞).
(c)xsinxonthereallineR.
(d)tan1xonthereallineR.
(29)A3*3realsymmetricmatrixMadmits(1,2,3)transposeand(1,1,1)transposeaseigenvectors.
ThetransposeofwhichofthefollowingissurelyaneigenvectorforMChooseallthecorrectoptions:(d)(a)(1,1,0)(b)(5,1,1)(c)(3,2,1)(d)noneoftheabove(30)Aninsectismovingalongthecurver=|cosθ|suchthatθ=πt/6,wheretistimemeasuredinseconds.
Whatisthedistancetravelledbytheinsectinthetimeintervalbetweent=1andt=2πππ/64